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Properties

Label 2-800-8.5-c5-0-8
Degree 22
Conductor 800800
Sign −0.451+0.892i-0.451 + 0.892i
Analytic cond. 128.307128.307
Root an. cond. 11.327211.3272
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 11.5i·3-s − 231.·7-s + 108.·9-s + 559. i·11-s − 107. i·13-s + 441.·17-s + 1.87e3i·19-s − 2.68e3i·21-s − 3.83e3·23-s + 4.07e3i·27-s + 3.36e3i·29-s + 7.95e3·31-s − 6.48e3·33-s + 1.06e4i·37-s + 1.25e3·39-s + ⋯
L(s)  = 1  + 0.743i·3-s − 1.78·7-s + 0.446·9-s + 1.39i·11-s − 0.177i·13-s + 0.370·17-s + 1.19i·19-s − 1.32i·21-s − 1.51·23-s + 1.07i·27-s + 0.744i·29-s + 1.48·31-s − 1.03·33-s + 1.28i·37-s + 0.131·39-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.451+0.892i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+5/2) L(s)=((−0.451+0.892i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.451+0.892i-0.451 + 0.892i
Analytic conductor: 128.307128.307
Root analytic conductor: 11.327211.3272
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ800(401,⋅)\chi_{800} (401, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :5/2), −0.451+0.892i)(2,\ 800,\ (\ :5/2),\ -0.451 + 0.892i)

Particular Values

L(3)L(3) ≈\approx 0.54642771340.5464277134
L(12)L(\frac12) ≈\approx 0.54642771340.5464277134
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
good3 1−11.5iT−243T2 1 - 11.5iT - 243T^{2}
7 1+231.T+1.68e4T2 1 + 231.T + 1.68e4T^{2}
11 1−559.iT−1.61e5T2 1 - 559. iT - 1.61e5T^{2}
13 1+107.iT−3.71e5T2 1 + 107. iT - 3.71e5T^{2}
17 1−441.T+1.41e6T2 1 - 441.T + 1.41e6T^{2}
19 1−1.87e3iT−2.47e6T2 1 - 1.87e3iT - 2.47e6T^{2}
23 1+3.83e3T+6.43e6T2 1 + 3.83e3T + 6.43e6T^{2}
29 1−3.36e3iT−2.05e7T2 1 - 3.36e3iT - 2.05e7T^{2}
31 1−7.95e3T+2.86e7T2 1 - 7.95e3T + 2.86e7T^{2}
37 1−1.06e4iT−6.93e7T2 1 - 1.06e4iT - 6.93e7T^{2}
41 1+9.96e3T+1.15e8T2 1 + 9.96e3T + 1.15e8T^{2}
43 1−925.iT−1.47e8T2 1 - 925. iT - 1.47e8T^{2}
47 1−8.06e3T+2.29e8T2 1 - 8.06e3T + 2.29e8T^{2}
53 1−7.95e3iT−4.18e8T2 1 - 7.95e3iT - 4.18e8T^{2}
59 1+1.68e4iT−7.14e8T2 1 + 1.68e4iT - 7.14e8T^{2}
61 1−1.12e4iT−8.44e8T2 1 - 1.12e4iT - 8.44e8T^{2}
67 1−3.36e4iT−1.35e9T2 1 - 3.36e4iT - 1.35e9T^{2}
71 1−8.86e3T+1.80e9T2 1 - 8.86e3T + 1.80e9T^{2}
73 1+5.55e4T+2.07e9T2 1 + 5.55e4T + 2.07e9T^{2}
79 1+6.94e4T+3.07e9T2 1 + 6.94e4T + 3.07e9T^{2}
83 1+1.02e4iT−3.93e9T2 1 + 1.02e4iT - 3.93e9T^{2}
89 1+9.24e4T+5.58e9T2 1 + 9.24e4T + 5.58e9T^{2}
97 1−8.86e4T+8.58e9T2 1 - 8.86e4T + 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.965327287991973077483317782101, −9.732486282206262698381751056529, −8.519702580257030735517287136212, −7.39165805646151480008715083274, −6.61900138018782737815354564497, −5.75703300798018064365204413912, −4.54211727578384964109809545914, −3.79943443174194161713384194470, −2.89262252142159593786964731452, −1.50972909546318885739284648934, 0.14084815648404601439808377300, 0.821467616039885812062343391732, 2.34487496336961606673315640316, 3.25752862759277821369152563139, 4.20516681787457169756255757504, 5.79535493991217841775337360206, 6.34840419425341870472603459783, 7.05068201299960722068627753270, 8.027474014460970399086944292357, 8.955392916343921477121183495162

Graph of the ZZ-function along the critical line